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Answer :
- The value of f(30) is 0, as the Taylor polynomial p(x) of degree 100 for the function f(x) about x = 3 is equal to 0 when x = 30.
1. Taylor polynomial p(x) = (x-3)² * 2! * (-1)ⁿ * 1 * (2-3)² * ... * n! * (x-3)¹⁰⁰ * 50!
2. When x = 30, p(30) = (30-3)² * 2! * (-1)ⁿ * 1 * (2-3)² * ... * n! * (30-3)¹⁰⁰ * 50!
3. Simplifying, we get p(30) = 27² * 2! * (-1)ⁿ * 1 * 0² * ... * n! * 27¹⁰⁰ * 50!
4. Since 0² = 0, p(30) = 0
In conclusion, the value of f(30) is 0, resulting from the simplification of the Taylor polynomial p(x) when x = 30. This highlights the importance of carefully examining the given function and applying the appropriate mathematical principles to obtain the correct answer.
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